Exam-Style Questions.

1.	IB Standard

The graph of \(f(x)=8-x^2\) crosses the x-axis at the points A and B.

(a) Find the x-coordinate of A and of B.

(b) The region enclosed by the graph of \(f\) and the x-axis is revolved 360^o about the x-axis. Find the volume of the solid formed.

Worked Solution

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2.	IB Standard

The acceleration, \(a\) ms^-2 , of an object at time \(t\) seconds is given by

$$a=\frac1t+4sin3t, (t\ge1)$$

The object is at rest when \(t=1\).

Find the velocity of the object when \(t=7\).

Worked Solution

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3.	IB Analysis and Approaches

(a) Find \(\int (4x+5) dx\).

(b) Given \(f'(x) = 4x+5\) find \(f(x)\) if \(f(3.4) = 10.12\).

Worked Solution

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4.	IB Analysis and Approaches

Given that \( \frac{dy}{dx} = \sin(x + \frac{\pi}{3})\) and \(y = 5\) when \(x = \frac{8\pi}{3}\), find \(y\) in terms of \(x\).

Worked Solution

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5.

A-Level

(a) Find \( \frac{dy}{dx} \) when:

$$ y= (7-5x^2)^{ \frac12 } $$

(b) Find the following integral:

$$ \int (1 - cos3x) dx$$

Worked Solution

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6.	IB Standard

Consider the graph of the function \(f(x)=x^2+2\).

(a) Find the area between the graph of \(f\) and the x-axis for \(2\le x \le 3\).

(b) If the area described above is rotated 360^o around the x-axis find the volume of the solid formed.

Worked Solution

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7.

A-Level

(a) Express the algebraic fraction

$$ \frac{6x^2 - 47x + 49}{(5-x)(1-2x)} $$

in the form

$$A + \frac{B}{5-x} + \frac{C}{1-2x} $$

where \(A\), \(B\) and \(C\) are integers.

(b) Hence show that the following integral equates to 3.03 correct to three significant figures.

$$ \int^{0.25}_0 \frac{6x^2 - 47x + 49}{(5-x)(1-2x)} dx $$

Worked Solution

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8.	IB Standard

Make a sketch of a graph showing the velocity (in \(ms^{-1}\)) against time of a particle travelling for six seconds according to the equation:

$$v=e^{\sin t}-1$$

(a) Find the point at which the graph crosses the \(t\) axis.

(b) How far does the particle travel during these first six seconds?

Worked Solution

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9.	IB Standard

Find the value of \(a\) if \(\pi \lt a \lt 2\pi\) and:

$$ \int_\pi^a sin3x dx = -\frac13$$

Worked Solution

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10.	IB Standard

This graph represents the function \(f:x\to a \cos x, a\in \mathbf N\)

(a) Find the value of \(a\).

(b) Find the area of the shaded region.

Worked Solution

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11.	IB Standard

Find \(f(x)\) if \(f'(x)=6 \sin2x\) and the graph of \(f(x)\) passes through the point \((\frac{\pi}{3},11)\).

Worked Solution

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12.	IB Analysis and Approaches

Consider the function \(f\) defined by \(f(x) = 25e^{x-5}\) for \(x \in \mathbb{R}^+\).

(a) Find the coordinates of the points where the graph of \(f\) intersects the line \(y=x\).

The line \(L\) has a gradient of \(-1\) and is a normal to the graph of \(f\) at the point \(R\).

(b) Find the exact coordinates of \(R\).

(c) Show that the equation of the line \(L\) is \(y=-x+6- \ln{25}\).

(d) Find the area of the region enclosed by the graph of \(f\) and its inverse.

Worked Solution

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13.	IB Standard

The following diagram shows the graph of \(f(x) = \cos(e^x) \; \text{for} \; 0 \le x \le 0.5\).

(a) Find the x-intercept of the graph of \(f(x)\).

The region enclosed by the graph of \(f(x)\), the y-axis and the x-axis is rotated 360° about the x-axis.

(b) Find the volume of the solid formed.

Worked Solution

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14.	IB Analysis and Approaches

Find \(f(x)\) if:

$$ f'(x) = \frac{12x}{\sqrt{3x^2+4}}$$

given that \(f(0) = 10 \)

Worked Solution

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15.	IB Analysis and Approaches

Let \(f(x) = \frac{ln3x}{kx} \) where \( x \gt 0\) and \( k \in \mathbf Q^+ \).

(a) Find an expression for the first derivative \(f'(x) \).

The graph of \(f\) has exactly one maximum point at P.

(b) Find the x-coordinate of P.

The graph of \(f\) has exactly one point of inflection at Q.

(c) Find the x-coordinate of Q.

(d) The region enclosed by the graph of \(f\), the x-axis, and the vertical lines through P and Q has an area of one square unit, find the value of \(k\).

Worked Solution

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